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Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving ||
Reasoning from Evidence
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Classroom Assessment Techniques
'Convincing and Proving' Tasks
(Screen 4 of 4)
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Variations
The tasks included in this site can be downloaded and used without modification. If you choose to develop your own "Convincing and Proving" task, you can follow the pattern used in these tools.
For the first type of task, where mathematical statements are to be categorized as "always, sometimes or never true," it is often helpful to choose statements which are in fact common misconceptions that are often regarded as "always true" when they are in fact only true over a limited domain. Thus, you could choose statements such as, "When you add two fractions, you simply add the tops and the bottoms" or, "If you double the circumference of a circle you double its area." Note that while it might be quite easy to quickly dismiss these as not being "always true," finding all the cases when they are "sometimes true" is demanding. Thus, all students, regardless of background and ability level, can be challenged.
Designing the second type of task is more difficult. These require students to evaluate given "proofs" and discover flaws in them. Designing plausible errors can be quite tricky. One possibility is to set the students a normal "proof task" and select some of their own responses to analyze. These responses could be typed, copied (with names removed) and circulated to the whole class. Everyone can then discuss and assess their worth. As it is very uncomfortable to have your ideas pulled apart by your peers, anonymity should be carefully preserved. There are many famous examples of fallacious proofs which students may enjoy looking at in books, such as Eugene Northrop's 'Riddles in Mathematics' (Pelican).
One sample 'proof' that 1 = 2 is shown below:
| Let
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x = y
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| Then
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xy = y2
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| and
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xy - x2 = y2 - x2
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| and
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x(y - x) = (y + x)(y - x)
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| Dividing by
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(y - x)
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| Yields
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x = y + x
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| But
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x = y
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| So
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x = 2x
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| Dividing by
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x
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| Yields
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1 = 2
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Analysis
In the first type of task, the students' work can be measured against three criteria:
- whether students can understand a statement and test it in particular cases;
- whether the student can identify suitable examples or counterexamples which will suggest the truth or falsehood of a given statement; and,
- whether the student can provide a convincing argument or 'proof' to support their conclusion.
In the second type of task, the students' work may be measured against two criteria:
- whether they are able to identify a correct proof and explain their choice; and,
- whether they can find mistakes or errors in logic within other given 'proofs'.
This generic scoring rubric may be modified and adapted for specific tasks.
Figure 1: Scoring rubric for "Always, sometimes, or never true" tasks
| Category of performance
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Typical response
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| The student needs significant instruction
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Student can understand a general statement, but cannot test it in specific cases.
Student can only follow a small part of the given proof and cannot even begin to evaluate it.
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| The student needs some instruction
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Student can understand a general statement, and can make up an example to test it. This example may not be well chosen and may lead the student to the wrong conclusion about the statement.
Student can follow a given proof. The student cannot begin to explain, however, why or where it is flawed.
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| The student's work needs to be revised
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Student can understand a general statement, and can test it with a suitable range of examples. The student draws an appropriate conclusion, but does not attempt to prove or justify it in general.
Student can follow a given proof, and can correctly see that it is flawed. The student makes a partial attempt to explain the nature of the error, but this explanation is unclear or has significant omissions.
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| The student's work meets the essential demands of the task
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Student can understand a general statement, and can test it with a suitable range of examples. The student draws an appropriate conclusion, and makes a good attempt to prove or justify it in general.
Student can follow a given proof, and can correctly see that it is flawed. The student makes a good attempt to explain the nature of the error.
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Tell me more about this technique:
Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving ||
Reasoning from Evidence
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